Array Signal Processing Algorithms for Beamforming and Direction Finding

Transcription

1 Array Signal Processing Algorithms for Beamforming and Direction Finding This thesis is submitted in partial fulfilment of the requirements for Doctor of Philosophy (Ph.D.) Lei Wang Communications Research Group Department of Electronics University of York December 2009

2 ABSTRACT Array processing is an area of study devoted to processing the signals received from an antenna array and extracting information of interest. It has played an important role in widespread applications like radar, sonar, and wireless communications. Numerous adaptive array processing algorithms have been reported in the literature in the last several decades. These algorithms, in a general view, exhibit a trade-off between performance and required computational complexity. In this thesis, we focus on the development of array processing algorithms in the application of beamforming and direction of arrival (DOA) estimation. In the beamformer design, we employ the constrained minimum variance (CMV) and the constrained constant modulus (CCM) criteria to propose full-rank and reduced-rank adaptive algorithms. Specifically, for the full-rank algorithms, we present two low-complexity adaptive step size mechanisms with the CCM criterion for the step size adaptation of the stochastic gradient (SG) algorithms. The convergence and steady-state properties are analysed. Then, the full-rank constrained conjugate gradient (CG) adaptive filtering algorithms are proposed according to the CMV and CCM criteria. We introduce a CG based weight vector to incorporate the constraint in the design criteria for solving the system of equations that arises from each design problem. The proposed algorithms avoid the covariance matrix inversion and provide a trade-off between the complexity and performance. In reduced-rank array processing, we present CMV and CCM reduced-rank schemes based on joint iterative optimization (JIO) of adaptive filters. This scheme consists a bank of full-rank adaptive filters that forms the transformation matrix, and an adaptive reduced-rank filter that operates at the output of the bank of filters. The transformation matrix and the reduced-rank weight vector are jointly optimized according to the CMV or CCM criteria. For the application of beamforming, we describe the JIO scheme for both the direct-form processor (DFP) and the generalized sidelobe canceller (GSC) structures. For each structure, we derive SG and recursive least squares (RLS) type algorithms to iteratively compute the transformation matrix and the reduced-rank weight vector for the reduced-rank scheme. An auxiliary vector filtering (AVF) algorithm based on the CCM design for robust beamforming is presented. The proposed beamformer decomposes the

3 adaptive filter into a constrained (reference vector filter) and an unconstrained (auxiliary vector filter) component. The weight vector is iterated by subtracting the scaling auxiliary vector from the reference vector. For the DOA estimation, the reduced-rank scheme with the minimum variance (MV) power spectral evaluation is introduced. A spatial smoothing (SS) technique is employed in the proposed method to improve the resolution. The proposed DOA estimation algorithms are suitable for large arrays and to deal with direction finding for a small number of snapshots, a large number of users, and without the exact information of the number of sources.

4 CONTENTS Acknowledgements viii Declaration ix Glossary List of Symbols List of Figures List of Tables x xii xii xvi 1. Introduction Overview Prior Work Beamforming DOA Estimation Contributions Thesis Outline Notation List of Publications

5 2. Adaptive Step Size CCM SG Algorithms for Adaptive Beamforming Introduction Array Structure Adaptive Array Structure Antenna Array and Direction of Arrival System Model Adaptive SG Algorithm Proposed Adaptive Step Size Mechanisms Modified Adaptive Step Size (MASS) Time Averaging Adaptive Step Size (TAASS) Computational Complexity Analysis of the Proposed Algorithms Convergence Analysis Sufficient Condition for the Convergence of the Mean Weight Vector 21 Steady-State Step Size Value for MASS Steady-State Step Size Value for TAASS Steady-state Analysis Tracking Analysis Simulation Results Conclusions

6 3. Constrained Adaptive Filtering Algorithms Based on the Conjugate Gradient Method for Beamforming Introduction Problem Statement Proposed Adaptive CG algorithms Conjugate Gradient Algorithms Proposed Conventional Conjugate Gradient (CCG) Algorithms.. 42 The CMV-CCG Algorithm The CCM-CCG Algorithm Proposed Modified Conjugate Gradient (MCG) Algorithms The Proposed CMV-MCG Algorithm The Proposed CCM-MCG Algorithm Analysis of the Proposed Methods Global Convergence and Properties Computational Complexity Convergence Analysis Simulations Conclusions Adaptive Reduced-rank CMV Beamforming and DOA Algorithms Based on Joint Iterative Optimization of Filters Introduction for Beamforming

7 4.2 Problem Statement Proposed Reduced-rank Method Adaptive Algorithms Stochastic Gradient Algorithm Recursive Least Squares Algorithms Complexity of Proposed Algorithms Automatic Rank Selection Analysis of Algorithms Stability Analysis MSE Convergence Analysis Simulations MSE Analysis Performance SINR Performance Introduction for DOA Estimation Problem Statement The JIO Scheme for DOA Estimation Proposed Reduced-Rank Algorithms Simulations Conclusions Adaptive Reduced-rank CCM Algorithms Based on Joint Iterative Optimization

8 of Filters and Auxiliary Vector Filtering for Beamforming Introduction Preliminary Works Full-rank Beamformer Design for the DFP Full-rank Beamformer Design for the GSC Reduced-rank Beamformer Design Beamformer Design for the DFP Beamformer Design for the GSC Proposed CCM Reduced-rank Scheme Proposed CCM Reduced-rank Scheme for the DFP Proposed CCM Reduced-rank Scheme for the GSC Adaptive Algorithms of the CCM Reduced-rank Scheme Stochastic Gradient Algorithms The SG Algorithm for the DFP The SG Algorithm for the GSC Recursive Least Squares Algorithms The RLS Algorithm for the DFP The RLS Algorithm for the GSC Gram-Schmidt Technique for Problem Automatic Rank Selection

9 5.6 Analysis of the Proposed Algorithms Complexity Analysis Analysis of the Optimization Problem Simulations Proposed CCM-AVF Algorithm Proposed CCM-AVF Scheme Proposed CCM-AVF Algorithm Interpretations about Proposed CCM-AVF Algorithm Simulations Conclusions Conclusions and Future Work Summary of Work Future Work Appendix 130 A. Derivation of (2.28) B. Convexity Condition for the CCM Criterion C. Preservation of MV and Existence of Multiple Solutions D. Analysis of the Optimization of the JIO CMV Scheme

10 E. Derivation of Transformation Matrix F. Derivation of (5.31) Bibliography

11 Acknowledgements I would like to express my most sincere gratitude to my supervisor, Dr. Rodrigo C. de Lamare, for his help, valuable supervision and useful advice for my research, without which much of this work would not have been possible. Further thanks go to all members of the Communications Research Group, for their help and support throughout my first year of research. Finally, my deep gratitude goes to my parents and my wife for their unconditional support, end-less love and encouragement. L. Wang, Ph.D. Thesis, Department of Electronics, University of York viii 2009

12 Declaration Some of the research presented in this thesis has resulted in some publications. These publications are listed at the end of Chapter 1. All work presented in this thesis as original is so, to the best knowledge of the author. References and acknowledgements to other researchers have been given as appropriate. L. Wang, Ph.D. Thesis, Department of Electronics, University of York ix 2009

13 Glossary AP ASS AVF BER BPSK CCG CCM CG CM CMV CRB CS DFP DOA EMSE ESPRIT FIR FR FSS GS GSC JIO LS MAI MASS MCG ML MMSE MSWF MUSIC MV PC RGS RLS RMSE RR SDMA Alternating Projection Adaptive Step Size Auxiliary Vector Filtering Bit Error Rate Binary Phase Shift Keying Conventional Conjugate Gradient Constrained Constant Modulus Conjugate Gradient Constant Modulus Constrained Minimum Variance Cramér-Rao bound Cross-Spectral Director Form Processor Direction of Arrival Excess Mean Square Error Estimation of Signal Parameters via Rotational Invariance Technique Finite Impulse Response Full-Rank Fixed Step Size Gram-Schmidt Generalized Sildelobe Canceller Joint Iterative Optimization Least Squares Multiple Access Interference Modified Adaptive Step Size Modified Conjugate Gradient Maximum Likelihood Minimum Mean Square Error Multistage Wiener Filter MUltiple SIgnal Classification Minimum Variance Principle Component Recursive Gram-Schmidt Recursive Least Squares Root Mean Square Error Reduced-Rank Space Division Multiple Access L. Wang, Ph.D. Thesis, Department of Electronics, University of York x 2009

14 SG SINR SNR SOI SS SVD TAASS ULA Stochastic Gradient Signal-plus-Interference-to-Noise Ratio Signal-to-Noise Ratio Signal of Interest Spatial Smoothing Singular Value Decomposition Time Averaging Adaptive Step Size Uniform Linear Array

15 List of Symbols Gradient O( ) Complexity E[ ] Expectation Infinity R( ) Real part C( ) Imaginary part Π Product Sum Euclidean norm Absolute trace( ) Trace of a matrix I Identity matrix span( ) Span a space T k proj( ) C m q R m q Chebyshev polynomial of degree k Projection operator space of complex valued matrices of size m by q space of real valued matrices of size m by q L. Wang, Ph.D. Thesis, Department of Electronics, University of York xii 2009

16 LIST OF FIGURES 2.1 An adaptive array structure A linear equally spaced array oriented along the axis receives a plane wave from direction (θ, φ) Adaptive ULA structure Simulation result for Case A, Expt. 1: Curves for BER versus input SNR for various adaptive algorithms Simulation result for Case A, Expt. 2: Curves for BER versus the number of snapshots for (a) ideal steering vector condition and (b) steering vector with mismatch= 1 o Simulation results for Case B, Expt.3: Curves for EMSE versus the number of snapshots for (a) MASS and (b) TAASS Simulation results for Case B, Expt.4: Curves for EMSE versus the number of snapshots for (a) MASS and (b) TAASS Simulation result for Case B, Expt. 5: Curves for EMSE versus input SNR for proposed adaptive algorithms Simulation result for Case B, Expt. 6: Curves for EMSE versus input SNR for proposed adaptive algorithms Simulation results for Case C, Expt. 7: Curves for output SINR versus the number of snapshots for various adaptive algorithms Simulation result for Case C, Expt. 7: Curves for step size values versus the number of snapshots for proposed adaptive algorithms

17 3.1 Output SINR versus the number of snapshots with q = 4 users and m = 10 sensor elements Output SINR versus input SNR for the proposed algorithms with q = 6 users and m = 16 sensor elements Output SINR versus the number of users (q) for the proposed algorithms with m = 16 sensor elements Array beampattern versus degree for the proposed algorithms with m = 16 sensor elements Output SINR versus the number of snapshots for (a) ideal steering vector condition. (b) steering vector with mismatch Output SINR versus the number of snapshots in a scenario where additional interferers suddenly enter and/or leave the system Step size values α and β of the proposed CCM-MCG algorithm in a scenario where additional interferers suddenly enter and/or leave the system The JIO reduced-rank structure MSE analytical versus simulated performance for the proposed reducedrank SG algorithm SINR performance of CMV algorithms against rank (r) with m = 32, SNR=15dB, N = 250 snapshots SINR performance of CMV algorithms against snapshots) with m = 32, SNR=15dB SINR performance of CMV (a) SG and (b) RLS algorithms against snapshots with m = 24, SNR= 12dB with automatic rank selection SINR performance of CMV algorithms against snapshots with m = 24, SNR= 12dB in a non-stationary scenario Probability of resolution versus input SNR

18 4.8 Probability of resolution versus input SNR RMSE versus input SNR (a) The full-rank DFP and (b) the full-rank GSC structures (a) The reduced-rank DFP and (b) the reduced-rank GSC structures Proposed reduced-rank scheme for (a) the DFP and (b) the GSC structures Complexity in terms of arithmetic operations versus the length of the filter m for the DFP structure Complexity in terms of arithmetic operations versus the length of the filter m for the GSC structure Output SINR versus input SNR with m = 32, q = 5, SNR= 10 db, (a) µ Tr = 0.002, µ w = 0.001, r = 5 for SG, µ Tr = 0.003, µ w = , r = 5 for GS; (b) α = 0.998, δ = δ = 0.03, r = 5 for RLS, α = 0.998, δ = δ = 0.028, r = 5 for RGS of the proposed CCM reduced-rank scheme Output SINR versus the number of snapshots with m = 32, q = 7, SNR= 10 db, µ Tr = 0.003, µ w = 0.003, r = 5 for SG, µ Tr = , µ w = 0.003, r = 5 for GS, α = 0.998, δ = δ = 0.025, r = 5 for RLS, α = 0.998, δ = δ = 0.02, r = 5 for RGS of the DFP structure Output SINR versus input SNR with m = 32, q = 7, SNR= 10 db, µ Tr = , µ wgsc = 0.002, r = 5 for SG, µ Tr = 0.003, µ wgsc = 0.003, r = 5 for GS, α = 0.998, δ = δ = 0.01, r = 5 for RLS, α = 0.998, δ = δ = , r = 5 for RGS of the GSC structure Output SINR versus rank r with m = 32, q = 7, SNR= 10 db Output SINR versus the number of snapshots with m = 32, q = 10, SNR= 10 db, (a) µ Tr = 0.003, µ w = for SG, µ Tr = 0.003, µ w = for GS; (b) α = 0.998, δ = δ = 0.03 for RLS, α = 0.998, δ = δ = 0.026, r = 5 for RGS with the automatic rank selection technique

19 5.11 Output SINR versus input SNR with m = 32, q 1 = 8, q 2 = 11, SNR= 10 db, µ Tr = 0.003, µ w = , r = 5 for SG, µ Tr = 0.003, µ w = 0.001, r = 5 for GS, α = 0.998, δ = δ = 0.033, r = 5 for RLS, α = 0.998, δ = δ = 0.028, r = 5 for RGS of the proposed CCM reduced-rank scheme Output SINR versus the number of snapshots for (a) ideal steering vector; (b) steering vector mismatch 1 o Output SINR versus the number of iterations

20 LIST OF TABLES 2.1 Simulation parameters for Case A and Case C Simulation parameters for Case B The CMV-CCG algorithm The CCM-CCG algorithm The CMV-MCG algorithm The CCM-MCG algorithm Comparison of the computational complexity Proposed JIO-RLS algorithm The JIO-CCM-SG algorithm for DFP The JIO-CCM-SG algorithm for GSC The JIO-CCM-RLS algorithm for DFP The JIO-CCM-RLS algorithm for GSC Computational complexity of algorithms for DFP Computational complexity of algorithms for GSC Proposed CCM-AVF algorithm

21 1. INTRODUCTION Contents 1.1 Overview Prior Work Contributions Thesis Outline Notation List of Publications Overview Array processing is an area of signal processing that has powerful tools for extracting information from signals collected using an array of sensors. The information of interest in the signal corresponds to either the content of the signal itself as often found in communications or the location of the source or reflection that produces the signal in radar and sonar systems [1]. These signals propagate spatially through a medium and the wavefront is captured by the sensor array. The sensor array data is processed to extract useful information. Some statistical and adaptive signal processing techniques, including parameter estimation and adaptive filtering (most related topics in the thesis), are extended to sensor array applications. Array processing finds numerous applications in wireless communications [2,3], radar [4], and sonar [5], and is a promising topic for emerging technologies such as wireless sensor networks [6]. Other applications include seismology, biomedicine, and imaging [7].

22 2 1.2 Prior Work Amongst the most interesting topics of array processing techniques are beamforming and the estimation of the direction of arrival (DOA) of signals, which are widely used in areas that include radar, sonar, acoustics, astronomy, seismology, and communications [8, 9]. Here, we focus on their developments in communications, more specifically, wireless communications, for attenuating interference, improving estimation accuracy, and locating the positions of the sources. To simplify the discussion, we concentrate on uniform linear array (ULA), which composes of a number of identical elements arranged in a single line with uniform spacing. The extension of this material to other array configurations is fairly straightforward in most cases [1] and will be considered as a topic for future investigation Beamforming Generally, an array captures spatially propagating signals arriving from a certain direction and processes them to obtain useful information. To this end, we intend to linearly combine the signals from all the sensors with coefficients in a manner, so as to estimate transmitted data radiating from a specific direction. This operation is known as beamforming [10, 11] since the weighting process emphasizes signals from a particular direction while attenuating those from other directions, which can be regarded as casting or forming a beam. In beamforming, an array processor steers a beam to a certain direction by computing a properly weighted sum of the individual sensor signals just as an finite impulse response (FIR) filter generates an output (at a frequency of interest) that is the weighted sum of time samples. It is convenient to view a beamformer as a frequencyselective filter. Thus, for the beamformer design, filtering techniques can be extended to sensor array applications [12]. According to weighting values are fixed or not, beamformers can be divided into conventional beamformers and adaptive beamformers [12]. Conventional beamformers employ a fixed set of weightings and time-delays to combine the signals from the sensors in the array, primarily using only information about the location of the signal of interest (SOI) relative to the sensor array. However, an array must contend with undesired signals arriving from other directions, which may prevent it from successfully extracting the SOI for which is was designed. Under this condition, the array needs to adjust its response of the received signals for rejecting unwanted signals from other directions. The resulting array is an adaptive array and the corresponding adaptive beamformer updates the weight-

23 3 ings by optimizing a certain criterion of performance (subject to various constraints). The adaptive beamformers have better resolution and interference rejection capability than the conventional one [13]. Much effort has been devoted over the past decades to devise adaptive beamformers [14]- [17]. The weighting values in adaptive beamforming are calculated according to optimization of certain criteria [8]. The most promising criteria employed are the constrained minimum variance (CMV) [14] and the constrained constant modulus (CCM) [8] due to their simplicity and effectiveness. The CMV criterion determines the weights by minimizing the beamformer output power, subject to the constraint that the response should be unity in the direction of the SOI. The CCM criterion is a positive measure of the deviation of the beamformer output from a constant modulus value subject to a constraint on the array response of the SOI. The CCM beamformer minimizes the square modulus of the deviation while retaining the gain along the look direction to be constant. Numerous adaptive filtering algorithms have been employed in beamforming to realize the beamformer design [14]- [18]. Among existing algorithms, the stochastic gradient (SG) is a low-complexity algorithm that employs instantaneous gradient values for iteratively computing weighting values. Its performance is acceptable in many applications. However, its convergence and steady-state behavior depend on the step size and eigenvalue spread of the input covariance matrix. When the eigenvalues are widely spread or/and inappropriate step size values are selected, convergence may be slow and other adaptive algorithms with better convergence characteristics should be considered, e.g., the recursive least squares (RLS) [18]. The RLS algorithm has fast convergence and is independent of the eigenvalue spread of the covariance matrix for stationary inputs. The key problems of this algorithm are high complexity and numerical instability. An alternative method is the CG [19]. It generates iteration (weight) vectors to approximate the optimum, residual vectors corresponding to the iterates, and direction vectors used for updating the iterates and residuals. The algorithm obtains the solution in several iterations. The CG algorithm represents a tradeoff between SG and RLS since it has a faster convergence rate than SG and usually requires lower computational cost when compared with RLS [20]. The adaptive algorithms reviewed above belong to a class that can be called full-rank array processing algorithms. Another class of algorithms that are attracting significant interest is that of reduced-rank algorithms [21]- [25]. For the application of beamforming, reduced-rank schemes project the data received from the sensor array onto a lower dimensional subspace, and calculate the reduced-rank weight vector by minimizing a certain criterion (subject to various constraints) within this subspace for estimating the transmitted

24 4 signals. The conventional reduced-rank algorithms include the principle component (PC) and cross-spectral (CS) methods [21]- [24] that rely on an estimate of the signal subspace achieved by the eigen-decomposition of the input data covariance matrix. One of the wellknown reduced-rank schemes is the multistage Wiener filter (MSWF) [24, 25]. The process observed by the Wiener filter is first decomposed by a sequence of orthogonal blocking matrices forming an analysis filterbank, whose output is shown to be a process which is characterized by a tridiagonal covariance matrix. The corresponding error-synthesis filterbank is realized by means of a nested chain of scalar Wiener filters. These Wiener filters can be interpreted as well to be a Gram-Schmidt (GS) orthogonalization which results in an error sequence for the successive stages of the decomposed Wiener filter. The MSWF was derived based on the minimum mean squared error (MMSE) criterion [26], and its extensions that utilize the CMV and CCM criteria were reported in [27, 28]. Another technique that resembles the MSWF is the auxiliary-vector filtering (AVF) [29,30], which utilizes an iterative procedure to compute the weighting values. Its extension to adaptive filtering has been studied in [31] DOA Estimation Another important use of array processing techniques is for DOA estimation, that is, given a spatially propagating signal, the determination of its angle of arrival at the array. For the beamformer design, the assumption taken was that the angle of the desired signal relative to the antenna array was known exactly by the beamformer. Commonly, this information is employed in the constraint to make the beamformer steer in this direction. In practice, the actual angle from which the signal arrives is not precisely known. The purpose of the DOA estimation is to determine this angle. There are many DOA estimation algorithms found in the literature, and some of them are described in [1,9,32]. Three main kinds of algorithms are reviewed here, namely, conventional [9,33], subspace-based [34]- [37], and maximum likelihood (ML) [38] methods. The conventional DOA estimation algorithms steer beams in all possible directions and look for peaks in the output power. The implementation is reasonably simple but always requires a large number of sensor elements to achieve high resolution. The Capon s method [33] is one of the conventional algorithms. It minimizes the output power of the undesired interference while maintaining a constant gain along the look direction according to the minimum variance (MV) criterion subject to a constraint on the array response. By computing and plotting the Capon s spectrum over the possible directions, the DOAs can be estimated by finding the peaks. The estimation accuracy of the Capon s method depends on the number of snapshots and the array size.

25 5 Compared with the Capon s algorithm, the subspace-based methods [34, 35] achieve better resolution by exploiting the signal subspace of the input covariance matrix. The well-known methods include the multiple signal classification (MUSIC) [34], the estimation of signal parameters via rotational invariance technique (ESPRIT) [35] reported in the 1980s, the auxiliary vector (AV) [36], and the conjugate gradient (CG) [37] algorithms proposed more recently. MUSIC and ESPRIT algorithms consider the eigen-structure of the input covariance matrix to decompose the observation space into a signal subspace and a corresponding orthogonal noise subspace. MUSIC scans the possible angle directions, plots the power spectrum by making use of this orthogonality, and locates the peaks that correspond to the angles of sources. ESPRIT could reduce the computational requirement without an exhaustive search through all possible directions and achieve a better resolution. It derives its advantages by requiring that the sensor array has a structure that can be decomposed into two equal-sized identical subarrays with the corresponding elements of the two subarrays displaced from each other by a fixed distance. Both MUSIC and ESPRIT suffer from correlated sources. The AV method is developed based on the orthogonality of an extended non-eigenvector signal subspace with the true signal subspace and the scanning vector itself. As the scanning vector drops in the signal subspace, the DOAs are determined by finding the collapse in the extended signal subspace. The CG method can be considered as an extended version of the AV method since it applies the residual vectors of the CG algorithm in place of the AV basis. The AV and CG algorithms exhibit high resolution of uncorrelated and correlated sources with a small number of snapshots and at low signal-to-noise ratio (SNR). The performance of the ML method [9] is superior to the Capon s and subspace-based methods, especially in low SNR conditions or with a small number of snapshots. It performs well in correlated signal conditions. An attractive approach to simplify the computational complexity is based on an iterative technique referred to as alternating projection (AP) [38], that transforms the multivariate nonlinear maximization problem into a sequence of much simpler one-dimensional maximization problems. 1.3 Contributions In this thesis, I focus on the development of the array processing algorithms for the applications of beamforming and DOA estimation. Note that the applications here assume a scenario with an antenna array at the receiver that is usually found in uplink channels in wireless communications.

26 6 In this thesis, we introduce a low-complexity beamformer design based on the SG algorithm that is equipped with variable step size mechanisms. We employ two adaptive step size mechanisms to adapt the beamformer weight vector for the estimation of the transmitted data. The beamformer design is based on the CCM criterion. The characteristics of the proposed adaptive step size SG algorithms are investigated. Another kind of novel algorithms is derived based on the CG technique. The proposed CG-based algorithms enforce the constraint of the CMV and CCM criteria in the system of equations without the matrix inversion and exhibit fast convergence with low-complexity. We often have to deal with requirements that imply large arrays. However, most fullrank array processing algorithms require a large amount of samples to reach the steadystate when the number of elements in the filter is large. In dynamic scenarios, filters with many elements usually fail or provide poor performance in tracking signals embedded in interference and noise. Reduced-rank techniques were originally motivated to provide a way out of this dilemma. We introduce reduced-rank schemes based on joint iterative optimization (JIO) of filters with the CMV and CCM criteria and compare them with existing reduced-rank methods to show their improved performance in the studied scenarios. Besides, we present a CCM-based AVF algorithm for robust adaptive beamforming. Note that the adaptive beamforming algorithms described here are DOA-based and constrained, which means that the DOA of the SOI is known beforehand by the receiver and the constraint in the design criterion is related to the corresponding array response. For the DOA estimation, we present a new reduced-rank algorithm based on the MV power spectral evaluation. It is suitable for DOA estimation with large arrays and can be applied to arbitrary array geometries. This algorithm is efficient for problems of direction finding with a large number of sources, and/or without exact information of the number of sources, and does not require the singular value decomposition (SVD). Specifically, the contributions of this thesis are as follows: Two low-complexity SG algorithms with adaptive step size mechanisms are proposed. The algorithms employ the CCM criterion for the beamformer design. A complexity comparison is provided to show their advantages over existing methods. The condition on the step size for the convergence of the mean weight vector is established. The mean and mean-squared values of the step size, in the steady-state condition, are calculated for computation of the excess mean squared error (EMSE) and tracking analysis. The EMSE here considers the effects of additive noise and multiple access interference (MAI) when multiple users are introduced in the system. The energy conservation relation developed in [40]

27 7 is exploited in the analysis to simplify the derivation of the recursion for the weight error energy. Simulation experiments are carried out for the stationary and non-stationary scenarios, highlighting the improved performance achieved by the proposed mechanisms in comparison with the fixed step size (FSS) [18] and adaptive step size (ASS) [42, 47] ones. Constrained CG adaptive filtering algorithms are proposed for beamforming and provide an attractive tradeoff between the complexity and the performance. The proposed algorithms are derived according to the CMV and CCM criteria. A CGbased weight vector strategy is created for enforcing the constraint and computing the weight expressions. The devised algorithms avoid the matrix inversion and exhibit fast convergence with low complexity. The complexity and the convexity properties of the CCM algorithms are studied, and the convergence analysis of the CG-based weight vector is derived. A robust reduced-rank scheme based on joint iterative optimization (JIO) of adaptive filters is presented for the beamformer design. This scheme is designed according to the MV criterion subject to the constraint on the array response of the SOI. It consists of a bank of full-rank adaptive filters that forms the transformation matrix, and an adaptive reduced-rank filter that operates at the output of the bank of filters to estimate the desired signal. The proposed reduced-rank scheme provides an iterative exchange of information between the transformation matrix and the reduced-rank weight vector. We derive SG and RLS type algorithms to compute the transformation matrix and the reduced-rank weight vector. An automatic rank selection technique according to the MV criterion is developed to determine the most adequate rank of the proposed methods. An analysis of the stability and the convergence properties is presented and semi-analytical expressions are given for predicting their performance. The JIO reduced-rank scheme is applied to the DOA estimation based on the MV power spectral evaluation. It is specific for the large arrays condition and can be extended to arbitrary array geometries. We present a constrained RLS algorithm to compute the transformation matrix and the reduced-rank weight vector for calculating the output power of each scanning angle. A spatial smoothing (SS) technique is employed in the LS based method to improve the probability of resolution with highly correlated sources.the proposed algorithms exhibit high resolution for dealing with direction finding with a large number of users and/or without the exact

28 8 information of the number of sources. Simulation are conducted to show their performance in different scenarios. Considering the fact that the CCM-based beamformer achieves better output performance than the CMV-based one for constant modulus constellations, we present a CCM-based JIO reduced-rank scheme. The transformation matrix and the reduced-rank weight vector are jointly optimized according to the CM criterion subject to different constraints. We describe the proposed scheme for both the direct-form processor (DFP) and the generalized sidelobe canceller (GSC) structures. For each structure, we derive the SG and RLS algorithms. The Gram-Schmidt (GS) technique is applied to reformulate the transformation matrix and improve performance. An automatic rank selection technique is developed according to the CM criterion. The complexity and convexity analyses of the proposed methods are carried out. An auxiliary vector filtering algorithm with the CCM criterion is introduced for beamforming. This algorithm utilizes an iterative way to compute the weight vector for estimating the transmitted signal. It provides an efficient way to deal with filters with a large number of elements and shows superior performance under severe scenarios, e.g., steering vector mismatch. 1.4 Thesis Outline The structure of the thesis is as follows: In Chapter 2, a system model and design criteria are introduced for beamforming and DOA estimation. Several assumptions are given to facilitate the system model for the development of the proposed algorithms. Based on this system model, low-complexity adaptive step size mechanisms are introduced for the update of the step size in the SG algorithm. The beamformer design according to the CCM criterion is detailed in this chapter. Characteristics of the proposed algorithms are given and analytical expressions are developed to predict their performance. In Chapter 3, the conventional CG algorithm is reviewed. We develop modified versions of the CG algorithm for beamforming according to the CMV and CCM

29 9 criteria. The convergence properties are analyzed. In Chapter 4, a reduced-rank scheme based on joint iterative optimization of filters with the CMV criterion is proposed for beamforming. The SG and RLS type algorithms are described. The properties of the proposed methods are analyzed. Besides, a DOA estimation algorithm based on the proposed reduced-rank scheme is developed. An SS technique is employed to increase the probability of resolution. In Chapter 5, a CCM-based reduced-rank scheme is introduced. Two structures are investigated for the realization of the reduced-rank scheme. The GS and automatic rank selection techniques are combined in the proposed method for further enhancing the performance. A CCM-based auxiliary vector filtering algorithm is described. It provides an iterative way to compute the weight vector and exhibits a good performance in the studied scenarios. In Chapter 6, conclusions and a discussion on possibilities for the future work are presented. 1.5 Notation In this thesis, small and capital boldface letters are used to denotes vectors and matrices, e.g., x and R, respectively. Elements of the vector and matrix are denoted as x j and R k,l. With no specific explanation, all vectors are column vectors. The symbol I denotes the identity matrix of appropriate dimensions, and the boldface 0 denotes either a zero vector or a zero matrix. The notation x denotes the Euclidean norm of a vector. The variable i is used as a time index, e.g., x(i) is the vector x at time instant i. R( ) and J( ) denote the real and imaginary components of a complex number, respectively. The symbol denotes complex conjugate (for scalars), ( ) T denotes transpose, and ( ) H Hermitian transpose. The symbol E[ ] denotes the statistical expectation operator. 1.6 List of Publications Some of the research presented in this thesis has been published, submitted, or will be submitted for publication at the time of submission of this thesis.

30 10 Journal Papers 1. L Wang, R. C. de Lamare, and Y. L. Cai, Low-complexity adaptive step size constrained constant modulus SG algorithms for adaptive beamforming, Elsevier Signal Processing, vol. 89, pp , Dec L Wang and R. C. de Lamare, Constrained adaptive filtering algorithms based on conjugate gradient techniques for beamforming, submitted to IET Signal Processing, R. C. de Lamare, L. Wang, and R. Fa, Adaptive reduced-rank LCMV beamforming algorithms based on joint iterative optimization of filters: design and analysis, (accepted) Elsevier Signal Processing, L. Wang, R. C. de Lamare, and M. Yukawa, Adaptive reduced-rank constrained constant modulus algorithms based on joint iterative optimization of filters for beamforming, (under review in the 2nd round) IEEE Trans. Signal Processing, L. Wang and R. C. de Lamare, Robust auxiliary vector filtering algorithm for adaptive beamforming, submitted to IEEE Trans. Signal Processing. 6. L. Wang, R. C. de Lamare, and M. Haardt, Reduced-rank DOA estimation algorithms based on joint iterative optimization, (under preparation) for IEEE Trans. Signal Processing. Conference Papers 1. L. Wang and R. C. de Lamare, Constrained constant modulus RLS-based blind adaptive beamforming algorithm for smart antennas, 4th International Symposium on Wireless Communication Systems 2007, pp , Oct L. Wang and R. C. de Lamare, A novel constrained adaptive algorithm using the conjugate gradient method for smart antennas, 41st Asilomar Conference on Signals, Systems and Computers, pp , Nov L. Wang, Y. L. Cai, and R. C. de Lamare, Low-complexity adaptive step size constrained constant modulus sg-based algorithms for blind adaptive beamforming, IEEE International Conference on Acoustics, Speech and Signal Processing, 2008, pp , Mar

31 11 4. L. Wang and R. C. de Lamare, A new approach to reduced-rank DOA estimation based on joint iterative subspace optimization and grid search, 16th International Conference on Digital Signal Processing, pp. 1-6, July L. Wang and R. C. de Lamare, Adaptive reduced-rank constrained constant modulus beamforming algorithms based on joint iterative optimization of filters, (will have been presented) 2009 IEEE Workshop on Statistical Signal Processing, L. Wang and R. C. de Lamare, Robust auxiliary vector filtering algorithm based on constrained constant modulus design for adaptive beamforming, submitted to 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, Mar L. Wang, R. C. de Lamare, and M. Haardt, Reduced-rank DOA estimation based on joint iterative subspace recursive optimization and grid search, submitted to 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, Mar

32 2. ADAPTIVE STEP SIZE CCM SG ALGORITHMS FOR ADAPTIVE BEAMFORMING Contents 2.1 Introduction Array Structure System Model Adaptive SG Algorithm Proposed Adaptive Step Size Mechanisms Analysis of the Proposed Algorithms Simulation Results Conclusions Introduction We categorize this chapter into two parts. The first part presents a system model of the antenna array at the receiver. We will see that the system structures applied to beamformer design and DOA estimation are similar. The main difference is that, for adaptive beamforming, weighting values are adapted with the received data. Based on the system model, we introduce the adaptive algorithms for beamforming in the second part. Adaptive beamforming is employed widely in communication systems, such as spatial-division multiple access (SDMA) systems [11, 44]. Numerous array processing algorithms haven been developed for the beamformer design [8, 45, 46]. The SG is a low-complexity method for iteratively computing the weight vector to generate the beamformer output. Its performance is sensitive to the step size, which has to be adjusted to make a compromise between fast convergence and small misadjustment. Adaptive step size (ASS) mechanisms [42], [47] were employed to circumvent these problems. However, they cannot yield both fast tracking as well as small misadjustment with simple implementation.

33 13 We propose two low-complexity adaptive step size mechanisms for use with SG algorithms designed according to the CCM criterion. The origin of these mechanisms can be traced back to the works [48] and [49], where the algorithms were proposed with the MMSE criterion. The algorithms according to the CMV criterion were reported in [50]. We extend these mechanisms to the CCM criterion in consideration of its superior performance over the CMV for constant modulus constellations. The rest of this chapter is organized as follows: the array structure is introduced in Section 2.2, and the system model is described in Section 2.3. The adaptive SG algorithm based on the CCM beamformer design is introduced in Section 2.4. Section 2.5 presents the developed mechanisms for the SG method. Section 2.6 is dedicated to the convergence, steady-state and tracking analyses of the proposed algorithms. Simulation results are provided and discussed in Section 2.7, and conclusions are drawn in Section Array Structure Adaptive Array Structure Adaptive array is an efficient antenna structure to realize beamforming, as shown in Fig This is an uniform linear array (ULA) that consists of m sensor elements. The sample received at each element is x j, where j = 1,..., m, and the corresponding weighting value is w j. The array processor processes the received samples to generate the output that is an estimate of the transmitted desired signal. The output signal passes through a feedback structure to get a reference signal, which is sent back to the adaptive processor for the weights update. The weight adaptation is performed by optimizing a cost function, which is also followed by the generation of the reference signal. Commonly, this reference is a deviation between the estimated desired signal and an assistant information, e.g., a constant modulus condition in CM criterion. It is important to let the adaptive array processing algorithm control the weights following the change of the received data and maximize the quality of the estimated desired signal. In working with antenna array, it is very convenient to make use of vector notation, i.e., x = [x 1,..., x m ] T and w = [w 1,..., w m ] T denote the received data vector and weight vector, respectively.

34 14 Downconvert to Baseband A/D x 1 w 1 Downconvert to Baseband A/D x 2 w 2 Output signal Demodulator Downconvert to Baseband A/D x m w m Generate Reference Signal Desired Signal Adaptive Algorithm Fig. 2.1: An adaptive array structure Antenna Array and Direction of Arrival The direction of arrival (DOA) of the source signal of the antenna array can be determined by two angles, horizontal and azimuthal, which is illustrated in Fig.2.2. It is clear that the direction of the incident wave can be located by the horizonal angle θ and the azimuthal angle φ together. Thus, the DOA is a function related to them, i.e., f(θ, φ). For simplicity, unless otherwise noted, it is assumed that the incident wave arrives at the array in the horizontal plane, φ = π/2, so that the azimuthal direction θ completely specifies the DOA. 2.3 System Model In Fig. 2.3, we describe a generic system model applied to beamforming and the DOA estimation. It is essential to make two assumptions [9]: The propagating signals are assumed to be produced by point sources; that is, the size of the source is small with respect to the distance between the source and the

35 15 z y x Fig. 2.2: A linear equally spaced array oriented along the axis receives a plane wave from direction (θ, φ). array at the receiver end. The sources are assumed to be in the far field, so that the spherically propagating wave can be reasonably approximated with a plane wave. Now, let us suppose that q narrowband signals impinge on a ULA of m(m q) sensor elements. The DOAs of the sources are θ 0,..., θ q 1. The received vector x(i) C m 1 at the ith snapshot (time instant) can be modeled as x(i) = A(θ)s(i) + n(i), i = 1,..., N, (2.1) where θ = [θ 0,..., θ q 1 ] T R q 1 contains the DOAs of the sources, A(θ) = [a(θ 0 ),..., a(θ q 1 )] C m q comprises the steering vectors a(θ k ) = [1, e 2πj ι λc cosθ k,..., e 2πj(m 1) ι λc cosθ k ] T C m 1, (k = 0,..., q 1), where λ c is the wavelength and ι (ι = λ c /2 in general) is the inter-element distance of the ULA. In order to avoid mathematical ambiguities, the steering vectors a(θ k ) are assumed to be linearly independent, s(i) C q 1 contains the source data, n(i) C m 1 is temporally and spatially white sensor noise, which is assumed to be a zero-mean and Gaussian process, and N is the observation size given in snapshots. The output is given by y(i) = w H (i)x(i), (2.2)